A late Ignacio Mieres try completed a stunning fight-back for Exeter as Sale threw away a 20-point lead.
The Sharks looked to be in control after Henry Thomas and Sam Tuitupou both went over.
Richard Baxter reduced the deficit for Exeter before a penalty try after the break closed the gap to three points.
Chiefs replacement Mieres then levelled the score at 23-23 with his second penalty before touching down to secure a dramatic victory.
As well as failing to add any points in the second half, the result also meant that Sale failed to move up to third in the Premiership table.
Nick MacLeod's two early penalties for Sale were followed by the game's opening try as Rob Miller broke through a clutch of Exeter defenders before laying the ball off to Thomas, who crashed over the line.
Tuitupou then latched onto Dwayne Peel's clever kick and he crossed the line before MacLeod added the extras for a second time.
Baxter then collected a quick pass from Gonzalo Camacho to score Exeter's first try before MacLeod's third penalty on the stroke of half-time gave Sale a 23-7 lead.
Gareth Steenson, who had converted Baxter's try, then kicked a penalty for the Chiefs before he was replaced by the highly influential Mieres.
With the hosts being regularly forced back towards their own line, Exeter earned a penalty try which was converted by the Argentine to leave the score finely poised at 23-20.
MacLeod then hit the post with a penalty before Mieres was successful with his own attempt between the posts at the opposite end to level the match.
And the Exeter replacement then raced clear late on before converting to secure a fine comeback for the Chiefs as Sale were forced to settle with a losing bonus-point.
Exeter head coach Rob Baxter: "The game looked to be getting away from us but credit to the guys.
"They got the try that stemmed the flow a bit in the first half, but really we were close to cracking.
"All we said at half-time was that we needed to do the things that re-established pressure.
"It was a massive challenge to our forwards but, to be fair, they cranked things up after half-time and forced some really good pressure at scrum time.
"We carried a lot better and got on the move a lot better and more accurately.
"We got the fundamentals right in the second half and that allowed us to play some pressure rugby."
Sale boss Steve Diamond: "We just didn't perform in the second half but full credit to Exeter.
"There was a lack of decisions going our way in the second half but you tend to get that if you're not being dominant.
"We talked at half-time about going out with the right mentality in the second half because it happened against Leicester and it happened again here.
"There are a lot of players who need to look at themselves because we've been beaten by a team on desire and decision-making.
"We go away with nothing - I'm embarrassed."
Sale: Miller; Brady, Leota, Tuitupou, Tuculet; MacLeod, Peel, Dickinson, Jones, Thomas; McKenzie, Myall; Gaskell, Seymour, Powell.
Replacements: Lavea for Miller (69), Tololima-Auva'a for McKenzie (74), Cobilas for Gaskell (58), Vernon for Powell (74).Not Used: Taylor, Buckley, Willis, Burrell.
Sin Bin: Thomas (54).
Exeter: Arscott, Camacho, Sestaret, Shoemark, Jess, Steenson, Phibbs, Sturgess, Clark, Tui, Hayes, Hanks, Johnson, Scaysbrook, Baxter.
Replacements: Mieres for Steenson (51), Thomas for Phibbs (40), Mitchell for Tui (57), White for Johnson (63).Not Used: Alcott, Moon, Phillips, Tatupu.
Ref: Tim Wigglesworth (RFU).